# Verification report — KOU-13.19 (Kourovka Notebook Problem 13.19, Yu. M. Gorchakov: regularity of quotients of subdirect products)

Verification date: 2026-10-18.

**Verdict.** The note gives a **complete answer: no in general**. Let Q be a subgroup of G_1 × ··· × G_n and H a
normal subgroup of Q such that Q and H both project onto every G_i (a "configuration of length n"). If Q/H is a
p-group, it is necessarily regular **if and only if n ≤ p** (Theorem 1.3).
- **Settled.** (a) n ≤ p: every p-group quotient is regular. (b) n ≥ p + 1: there are configurations whose quotient
  is an irregular finite p-group of class exactly n − 1, with pairwise isomorphic finite p-group factors, and also
  with free factors of finite rank; for n = p + 1 the quotient can be C_p ≀ C_p. The first case is p = 2, n = 3;
  an explicit example has |Q| = 512, |G_i| = 32 (extraspecial), Q/H ≅ D_8 (Proposition 6.3).
- **Not new.** Part (a): the class bound γ_n(Q) ≤ H is Gorchakov's lemma (Mat. Sb. 1965, Lemma 1, as cited in
  Gorchakov 2009 and in Kearnes–Mayr–Ruškuc 2018; see also Gorchakov's 1978 book, Proposition 4.7, as cited by
  Khukhro 1982); the same argument is in Khukhro (1982) and in Kearnes–Mayr–Ruškuc (2018, Theorem 2.1). That
  p-groups of class < p are regular is P. Hall's theorem (1934). Magnus's theorem on free groups (1937) and
  Jennings's theorem (1941) are classical. For p = 2 an irregular quotient with four perfect factors can be
  extracted from Kearnes–Mayr–Ruškuc (Theorem 4.2 with Lemma 6.1), which does not mention regularity or the
  Notebook; possibly also from examples of D. M. Evans (1993), via a lemma of Hrushovski (not determined).
- **New.** The threshold n ≤ p for every prime p, in particular for the odd primes; the realization theorems
  (Theorem 4.2: every nilpotent group of class ≤ n − 1 is a quotient with free factors; Theorem 5.1: every group P
  with D_n(P; Z) = 1, e.g. a finite p-group with D_n(P; F_p) = 1, is a quotient with isomorphic factors inside
  P^(2^(n−1) − 1)); the explicit examples.
- **Open** (not part of the question). Whether every finite p-group of class ≤ n − 1 is a quotient with finite
  factors; the smallest order of the factors (for p = 2, n = 3: not 8, at most 32); irregular quotients with perfect
  factors for odd p.

The note is unrefereed.

## Statement checked
- **Primary source.** E. I. Khukhro, V. D. Mazurov (eds.), The Kourovka Notebook, No. 21, arXiv:1401.0300v48
  (6 October 2026), Problem 13.19 (13th issue, 1995), proposed by Yu. M. Gorchakov. The text (local copy,
  lines 2529–2532 of the text extraction) assumes that Q and its normal subgroup H are subgroups of
  G_1 × ··· × G_n whose projections onto every G_i coincide with G_i, and asks: "If Q/H is a p-group, is Q/H a
  regular p-group?". No comment and no star follow; the next item is 13.20. No finiteness is assumed anywhere.
- **Related Archive entry.** Problem 7.8 (Gorchakov): does the nilpotency class of G/H increase with n? Answered
  "Yes" with a reference to Khukhro, Sibirsk. Mat. Zh. 23 (1982), no. 6, 178–180. (7.8 asks about growth; the upper
  bound is Gorchakov's 1965 lemma.)
- **Gorchakov's restatement.** Yu. M. Gorchakov, Sibirsk. Mat. Zh. 50 (2009), no. 4, 800–805 (English:
  Siberian Math. J. 50 (2009), 632–636). Read in the Russian original (EMIS copy). p. 800: the class bound is
  credited to [5, Lemma 1] = Mat. Sb. 67(109) (1965) 244–254. p. 801: Theorem (for every n there are G_1, …, G_n
  and subdirect products Q, H with Q/H nilpotent of class exactly n − 1), followed by the sentence stating the
  conjecture that such a Q/H is regular whenever it is a p-group. The construction: p odd, G = (C_p)^(p−1) ⋊ C_p
  with a unipotent Jordan block, p copies (so n = p, class p − 1: regular).
- **Corpus record.** ulamai/UnsolvedMath, KOU-13.19 (status `open`); statement identical to the Notebook text.
- **Definition used.** P. Hall's regularity: for all a, b there is c ∈ ℧_1(⟨a,b⟩′) with (ab)^p = a^p b^p c; used
  for all p-groups (finite or not). The negative answers use finite groups only; the positive answer needs no
  finiteness.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Literal question (all groups, any n, any p) | no; regular for all configurations iff n ≤ p | Theorem 1.3 |
| G_i required finite, or finite p-groups, or pairwise isomorphic | same answer | Theorem 5.1, Proposition 6.3, Example 6.4 |
| Gorchakov's 2009 conjecture (quotients of class exactly n − 1) | true exactly for n ≤ p; false for every n ≥ p + 1 | Theorem 1.3(b) (class exactly n − 1) |
| Q/H an infinite p-group | the positive half holds verbatim; the negative half is witnessed by finite quotients | Corollary 3.2 |
| n = 1, 2 | trivial, resp. abelian quotient: regular | Section 3 |
| G_i perfect | for p = 2 an irregular quotient exists (implicit in Kearnes–Mayr–Ruškuc); odd p not examined | Remark 7.1 |

## Results in the paper
- **Lemmas 2.1–2.3.** Hall–Petrescu formula (cited); p-groups of class < p are regular (Hall; proof given); a
  2-group of class ≤ 2 is regular iff abelian, and P/γ_3(P) is irregular for a finite non-abelian 2-group P.
- **Lemma 2.4.** Q = M_i H for M_i = Q ∩ ker π_i; enlarging H keeps a configuration.
- **Section 2.3.** Dimension subgroups: (D1)–(D4); Magnus's theorem (cited) and its consequence D_n(F; Z) = γ_n(F)
  for free F (Corollary 2.6); Jennings's theorem (cited).
- **Theorem 3.1 (Gorchakov).** γ_n(Q) ≤ H. **Corollary 3.2.** n ≤ p ⇒ every p-group quotient is regular.
- **Lemma 4.1.** Inclusion–exclusion: ω(w) = Σ_S (−1)^(n−|S|) ρ_{[n]∖S}(w) lies in Δ_Z(F)^n.
- **Theorem 4.2.** Free factors: every nilpotent P of class ≤ n − 1 generated by Y is a quotient with G_i free of
  rank (n − 1)|Y|; so the quotients of configurations of length n are exactly the nilpotent groups of class ≤ n − 1.
- **Theorem 5.1.** Finite factors: if D_n(P; Z) = 1 (e.g. a finite p-group with D_n(P; F_p) = 1), P is a quotient
  with Q = L ≤ P^(2^n − 1) and pairwise isomorphic G_i ≤ P^(2^(n−1) − 1). Remark 5.2: relation to the construction
  of Kearnes–Mayr–Ruškuc; D_n(P; F_p) = 1 is not necessary; what the construction gives without hypothesis.
- **Lemma 6.1.** W_p = C_p ≀ C_p: order p^(p+1), γ_(k+1) = (x − 1)^k B, class p, exponent p^2, generated by a and b_0,
  not regular, D_(p+1)(W_p; F_p) = 1 (Jennings), smallest possible order of an irregular p-group; W_2 = D_8.
- **Lemma 6.2.** UT_n(F_p): D_n(UT_n(F_p); Z) = 1, class exactly n − 1; W_p × UT_n(F_p) is irregular of class n − 1
  with D_n = 1 for n ≥ p + 1.
- **Proof of Theorem 1.3.** (a) Corollary 3.2; (b) Theorems 5.1 and 4.2 applied to P = W_p (n = p + 1) or
  W_p × UT_n(F_p) (n ≥ p + 2).
- **Proposition 6.3.** The example of order 512 with complete proof; the G_i are isomorphic and extraspecial.
- **Example 6.4.** Theorem 5.1 for (D_8, 3) (|L| = 2^12, |G_i| = 2^7) and (C_3 ≀ C_3, 4) (|L| = 3^32, |H| = 3^28,
  |K_i| = 3^13, |G_i| = 3^19), checked by exact group orders; negative controls (D_8, 2), (C_3 ≀ C_3, 3): |Z| = 2, 3.
- **Remarks 7.1–7.3.** The irregular quotient implicit in Kearnes–Mayr–Ruškuc; Hrushovski's lemma contains
  Gorchakov's setting; small factors (every G_i non-abelian if n = 3 and Q/H non-abelian; no example with three
  factors of order 8; the minimal order is open) and two further open questions.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program of the note** (`reproducibility/check_note.py`, Python standard library, 8.8 s, ALL CHECKS PASSED;
  re-run identically up to timing lines): (A) Proposition 6.3 exhaustively (cocycle identity on 64^3 triples, ψ on
  512^2 pairs, A_i, trivial intersection, |G_i| = 32 extraspecial with 19 involutions, |H| = 64, HA_i = Q, D_8
  irregular: Hall's identity fails for 24 of 64 pairs); (B) W_p for p = 2, 3, 5 with the values of Jennings's
  formula, and for p = 2, 3 the dimension subgroups computed directly in F_p W_p (orders 8, 2, 1 and 81, 9, 3, 1);
  (C) Example 6.4 by a deterministic Schreier–Sims algorithm (tested against enumeration), with the negative
  controls; (D) Lemma 4.1 pushed into F_p P on 400 random words each for (D_8, 3) and (C_3 ≀ C_3, 4); (E) small
  unitriangular groups (class m − 1; D_m = 1 for orders ≤ 125); (F) all subgroups of D_8^3, D_8^2 × Q_8,
  D_8 × Q_8^2, Q_8^3 (55,338, 22,746, 13,354, 9,434), 16,594, 13,010, 16,402, 28,306 configurations, all with
  abelian quotient and γ_3(Q) ≤ H.
- **Programs with which the results were first obtained** (not in the package): the example of order 512 (all
  Cayley edges), a larger example with |Q| = 2^15, Theorem 5.1 for (D_8, 3) by enumeration (|L| = 4096), dimension
  subgroups of D_8, Q_8, C_3 ≀ C_3 by linear algebra, and all 55,338 subgroups of D_8^3.
- **Verification run A** (own programs, partly with sympy): the example of order 512 with a different encoding
  (all 512^2 products; Q/H computed as a quotient: D_8, Hall's identity fails for 24 pairs); Theorem 5.1 for
  (D_8, 3), (Q_8, 3), (D_8, 4), (C_3 ≀ C_3, 4) by Schreier–Sims orders (sympy), negative control (C_3 ≀ C_3, 3) with
  |Z| = 3; W_p for p = 2, 3, 5 (orders, classes, exponents, witness, Jennings values; p = 2, 3 also in the group
  algebra). All checks passed.
- **Verification run B** (own toolkit with its own Schreier–Sims, cross-checked with sympy): an independent example
  of order 512 of its own design (generators of order 4, G_i extraspecial) and the example of the note; Theorem 5.1
  for (D_8, 3), (Q_8, 3), (D_8, 4), (C_3 ≀ C_3, 4) with negative controls (D_8, 2), (Q_8, 2), (C_3 ≀ C_3, 3) and two
  wrong inputs; dimension subgroups in group algebras = Jennings's formula (D_8, Q_8, C_3 ≀ C_3; C_5 ≀ C_5 by the
  formula); exhaustive searches for all D_8/Q_8 products (numbers as above); random and constructed tests of the
  class bound (about 14,600 configurations, n = 3, 4, no violation; the bound n − 1 reached in constructed cases);
  a search for factors of order 16 restricted to a class-2 model (none found; not exhaustive); re-runs of the original programs
  (identical outputs, md5 of the scripts as recorded).

## Independent verification runs
The results were first obtained with written proofs and programs. Two independent verification runs, both
AI-assisted, followed (2026-10-17/18); each re-derived the proofs and wrote its own programs, and neither read
the other's report.

| Item | Run A | Run B |
|---|---|---|
| Statement against the Notebook (v48) and the corpus record; Gorchakov 2009 read in Russian | CONFIRMED | CONFIRMED |
| Theorem 3.1 (class bound) and its credit | CONFIRMED (re-derived; credit checked in Gorchakov 2009, Kearnes–Mayr–Ruškuc, Khukhro) | CONFIRMED |
| Corollary 3.2 (Hall's criterion via Hall–Petrescu) | CONFIRMED; textbook theorem numbers not checked | CONFIRMED |
| Free factors (Magnus) | CONFIRMED (modulo Magnus's theorem) | CONFIRMED |
| Finite factors, inclusion–exclusion | CONFIRMED | CONFIRMED |
| C_p ≀ C_p irregular, D_(p+1) = 1 | CONFIRMED (proof; computation p = 2, 3, 5) | CONFIRMED (proof; computation) |
| Example of order 512 | CONFIRMED (proof and own computation) | CONFIRMED (own computation; second example) |
| p = 3, n = 4 configuration | CONFIRMED by exact orders | CONFIRMED by exact orders |
| Remark on Kearnes–Mayr–Ruškuc (p = 2, n = 4) | CONFIRMED from the arXiv source | CONFIRMED |
| Novelty | not answered in print as far as checked; implicit for p = 2 | same; Hrushovski–Evans added |
| Overall | CONFIRMED (complete dichotomy) | CONFIRMED |

Neither run found a mathematical error.

**Corrections required by the runs**, all applied in the note:
1. (A) Textbook theorem numbers quoted from memory (Huppert, Magnus–Karrass–Solitar, Serre, Dixon–du Sautoy–
   Mann–Segal) are not used; results are cited by name and section: Huppert Kap. III §9 (Hall–Petrescu) and §10,
   P. Hall 1934 §4, Magnus 1937 / Witt 1937, Jennings 1941.
2. (A) Kearnes–Mayr–Ruškuc: arXiv version 4 of 24 September 2020 (with the addendum) is the version cited, with
   DOI 10.1112/blms.12196 and Zbl 1441.20025; theorem numbers are those of arXiv version 4.
3. (A) The search for factors of order 8 is stated exactly: all D_8/Q_8 products for n = 3 (the program of the
   note repeats it); with the observation that every factor must be non-abelian this excludes three factors of
   order 8; the minimal order of the factors is stated as open.
4. (A) "The same argument" (not "exactly the proof"); Khukhro 1982 and Gorchakov 1978, Proposition 4.7, are
   credited.
5. (A) Description of A_i with the symmetry of (ε_ij) made explicit; the proof of the class bound is written at
   the level of subgroups.
6. (A) The counterexamples have class exactly n − 1 (for every n ≥ p + 1); the p = 3, n = 4 instance is certified
   by exact group orders.
7. (B) Hrushovski's lemma as quoted in the review of Evans (1993; DOI 10.1007/978-94-011-2080-7_7,
   Zbl 0847.20012) contains Gorchakov's setting; Evans's examples give non-abelian nilpotent quotients; whether a
   2-group arises was not determined (Remark 7.2, introduction, Scope and priority).
8. (B) Archive 7.8 asks whether the class grows with n; the upper bound is Gorchakov 1965, Lemma 1.
9. (B) Remark 5.2(2): "D_n(P; F_p) = 1 is not necessary" (the abelian n = 2 example does not bear on D_n(P; Z) = 1).
10. (B) C_p ≀ C_p is "an irregular p-group of the smallest possible order p^(p+1)".
11. (B) Identifiers: DOIs (checked with Crossref), Zbl numbers, Math-Net.Ru smj6667 for Khukhro 1982.

**Added when the note was written, after runs A and B.** Lemma 2.3 (2-groups of class ≤ 2); Lemma 6.2 and
the choice P = W_p × UT_n(F_p) for n ≥ p + 2 (class exactly n − 1 for every n ≥ p + 1); the derivation of the
free-factor theorem from Lemma 4.1 and Corollary 2.6 (instead of a separate power-series argument); the proof
that the factors of Proposition 6.3 are isomorphic and extraspecial (observed computationally by run B); the
program `check_note.py`. These items were examined by the second-stage run described next.

## Second-stage verification run (the final text), 2026-10-18

A further AI-assisted verification run examined the final text of the note, the items just listed first, and
the release package; its programs and recorded outputs are in `reproducibility/independent_run_2/`. It did not
use the code of the earlier runs or of the note.

- **Statement and sources.** The Notebook text (v48, lines 2529–2532) and the PDF page; the corpus record;
  Gorchakov 2009 in Russian (p. 800: the class bound credited to "[5, лемма 1]", [5] = Mat. Sb. 67 (1965) 244–254;
  p. 801: the theorem and the conjecture sentence; Lemma 1 passing to all n); the arXiv source of
  Kearnes–Mayr–Ruškuc (Theorem 2.1 and its proof; the definition of Γ_k, every Γ_k containing Δ_∅(G); Lemmas 3.3,
  4.1, Theorem 4.2; Section 6 with N block unitriangular over F_4 of exponent dividing 4 and Lemma 6.1(iv);
  Section 7 in outline, Lemma 7.1(iv); the addendum); the scan of Khukhro 1982 (class bound citing
  "[1, предложение 4.7]", Problem 7.8, Lie algebras over F_p with p > n, groups of exponent p); the zbMATH review
  of Evans 1993 (Hrushovski's lemma as quoted; finite non-abelian G_i of class < n/2; no prime named). All
  statements of the note about these sources are accurate; the extraction of the p = 2, n = 4 example from
  Kearnes–Mayr–Ruškuc was re-derived from their statements.
- **Proofs.** Every lemma, theorem, proposition and remark re-derived line by line, in particular Lemma 2.3,
  Lemma 6.2 (the commutator [1+E_1k, 1+E_k,k+1] = 1+E_1,k+1 re-derived with E_ij E_jl = E_il; the class and D_n(.; Z) = 1 of
  W_p × UT_n(F_p)), the derivation of Theorem 4.2 from Lemma 4.1 and Corollary 2.6 (ω(w) = w − 1 for w in all N_i),
  the isomorphism and the extraspecial structure of the factors of Proposition 6.3 (the induced form has Gram
  matrix of rank 4 on ⟨u_2, v_2, u_3, v_3⟩ with u_1, v_1 in its radical). No error was found.
- **Own computations** (eight programs; all checks pass): Proposition 6.3 in a different encoding with the
  factors and Q/H built as quotient groups and Hall's definition tested on all pairs; Lemma 2.3 on all subgroups
  of D_8 × D_8 and D_8 × Q_8 and on eight further 2-groups; Lemma 6.1 for p ≤ 5 in a permutation model (exhaustive
  Hall test for p = 2, 3; dimension subgroups by linear algebra for p = 2, 3; Jennings's formula for p ≤ 5);
  Lemma 6.2 by matrix computations, lower central series and group algebras (D_n(UT_n(F_p); F_p) = 1 for orders
  ≤ 729; W_2 × UT_4(F_2): class exactly 3, irregular, D_4 = 1 in F_2P; W_2 × UT_5(F_2): class exactly 4);
  Lemma 4.1 in the free group through the Magnus embedding (exact series over Z, random words, n ≤ 4) and
  exactly in Z[D_8], Z[Q_8], Z[C_4]; Theorem 5.1 by exact orders with sympy's Schreier–Sims for (D_8, 3), (Q_8, 3),
  (D_8, 4), (C_3 ≀ C_3, 4), (Heis(F_3), 3), (C_4, 2), (C_9, 2) and (W_2 × UT_4(F_2), 4) — the last a configuration
  of length 4 of finite 2-groups with |Q| = 2^64, |G_i| = 2^40, |H| = 2^55 and an irregular quotient of class
  exactly 3 — with negative controls (D_8, 2), (Q_8, 2), (C_3 ≀ C_3, 3), (W_2 × UT_4(F_2), 3); the searches of
  Remark 7.3 with an own enumerator (identical numbers; every quotient abelian); Theorem 3.1 on 360 random
  configurations and Corollary 3.2 on the p-group quotients among them with n ≤ p.
- **Package re-run.** `source.zip` extracted in a fresh folder; `python3 -I check_note.py` reproduced
  `check_note_output.txt` up to the timing lines (9.3 s). The program tests what Section 8 of the note says.
- **Literature** (2026-10-18; anonymous, paced, logged): arXiv API (five queries, including the 24 newest
  titles with "subdirect" in math.GR), Crossref (the eight DOIs of the reference list resolve to the cited works;
  no citing works of Gorchakov 2009, Kearnes–Mayr–Ruškuc or Evans 1993), zbMATH Open (no citing works; topic
  queries), OpenAlex (Kearnes–Mayr–Ruškuc: one citing work, list refused again by rate limiting), Semantic
  Scholar (no citing works), one web search. A July 2026 arXiv preprint with Lean-verified solutions to eight
  Notebook problems (van Doorn, Judin, Monticone, Morrison, arXiv:2607.17477) was found; its abstract lists the
  eight problems, none of which is 13.19. Verdict unchanged: no printed answer; p = 2 implicit in
  Kearnes–Mayr–Ruškuc and possibly in Evans.
- **Presentation.** All eleven pages rendered and read; house style, abstract, credit and scope statements
  checked; no overfull boxes.
- **Corrections required and made.** (1) The Verification paragraph records this run and no longer says that the
  items added at the writing stage are unexamined. (2) Section 8, item (E): the program checks Lemma 6.2(b) and
  D_n(R_n; F_p) = 1 (which implies (a)) for the groups of order ≤ 125. (3) The search paragraph names the queries
  of this run (OpenAlex added; seven web searches) and the July 2026 preprint. The package was rebuilt with
  `independent_run_2/` and this report.

## Relation to the literature, novelty and scope
- **Read.** Gorchakov 2009 (Russian original, EMIS) and Kearnes–Mayr–Ruškuc (arXiv version 4, LaTeX source)
  completely; Khukhro 1982 (Math-Net.Ru scan; text layer partly garbled, key passages legible); of Evans 1993 only
  the zbMATH review. Gorchakov 1965 and 1978 were not available; what the note says about them is taken from the
  citations in Gorchakov 2009, Kearnes–Mayr–Ruškuc and Khukhro. Hall 1934, Magnus 1937, Witt 1937, Jennings 1941
  and Huppert's book were not consulted; they are cited for classical statements at the level of sections.
- **Searches** (17–18 October 2026): arXiv API, zbMATH Open (Gorchakov's publications; works citing Gorchakov
  2009 and Kearnes–Mayr–Ruškuc; subdirect products with regular p-groups or nilpotency class), Crossref,
  Math-Net.Ru, EuDML, EMIS, Semantic Scholar, and six web searches. zbMATH, Crossref and Semantic Scholar list no
  work citing Gorchakov 2009 or Kearnes–Mayr–Ruškuc; OpenAlex counts one work citing Kearnes–Mayr–Ruškuc, which
  could not be identified (OpenAlex refused the requests: rate limiting). The 21st edition of the Notebook (2026)
  lists 13.19 without comment.
- **Novelty.** Apart from the implicit examples (Kearnes–Mayr–Ruškuc for p = 2, n = 4; possibly Evans for p = 2),
  no printed answer was found; a search that finds nothing is not a proof of novelty, and no priority is claimed.
  For odd primes no irregular quotient was found in print (Gorchakov's and Khukhro's examples are regular; the
  examples of Kearnes–Mayr–Ruškuc are 2-groups).
- **Scope.** The note answers Problem 13.19 completely (regular iff n ≤ p) and refutes Gorchakov's 2009 conjecture
  for every n ≥ p + 1. It does not determine the minimal order of the factors, nor whether every finite p-group of
  class ≤ n − 1 is a quotient with finite factors.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/